DOCSLIB.ORG

The Fourth Dimension (And More!)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Fourth Dimension (And More!) THE FOURTH DIMENSION (AND MORE!) The text leads us from zero dimensions (a point), to one dimension (a line), to two dimensions (a plane), to three dimensions (our familiar space), and then to four dimensions by “inking up” one level and then dragging it to create the next higher level. We approach the same problem by discussing “fundamental directions” and “coordinates.” Fundamental Directions and Dimension We introduce the idea of fundamental directions and use it to informally define dimension. As the book does, we develop our ideas by building up from a point. Imagine that we have a car that can move forward and in reverse. However, the steering mechanism has the odd property that it can only point the car in a few (discrete) directions. First, suppose that the steering mechanism is so ill-behaved that you cannot move the car at all. Then you are stuck at a single point and cannot travel anywhere. There are zero directions in which you can travel and the point forms a zero dimensional space (the word “space” is used since these ideas are explored in detail in the area of linear algebra when addressing the topic of vector spaces). Next, suppose that the steering mechanism will let you point the car in one direction. Then from wherever you start, you can reach any point on a line in the direction in which you can point the car (remember, you can drive both forward and backward). So there is one direction in which you can move and so you are stuck in a one dimensional space. This is why lines are said to be one dimensional. Next, suppose that you can point the car in exactly two different directions (but not opposite directions). In this case, you can reach any point in a plane by driving first in one direction, and then in the second direction: 1 Question. In the above pictures, suppose the blue direction is directly to the right and the red direction is to the upper right (at a 45◦ angle to the blue direction). How could we move directly up 10 units? How could we move 5 units to the left and 10 units up? Questions like this are easier to answer if the two directions are perpendicular (say left-right and up-down). Such a pair of fundamental directions is said to be orthogonal. Next, suppose our imaginary car can travel in three different directions. If all three directions lie in the same plane (i.e., they are coplanar), then we have the same situation as above and the car can travel to any point in the plane. So one of these three directions is redundant and we could make the same trips using only two directions. For example, suppose the car can move left-right, up-down, and upper right-lower left: Then any point in the plane can be reached by a combination of the orthogonal directions left- right and up-down. In particular, the upper right-lower left direction is a combination of equal parts of left-right and up-down.√ If you wanted to travel√ 10 units to the upper right, say, then you could do so by moving 5 2 units to the right and 5 2 units up. The upper right-lower left direction is said to be linearly dependent on the left-right and up-down directions. However, we have nothing special against upper right-lower left! We could also omit the up-down direction and could still get anywhere in the plane, as described above. So to travel anywhere in the plane, we must have the ability to move in two different “fundamental directions.” Adding more directions in the plane just adds redundancy (linear dependence). So, again, a plane is two dimensional. Question. Using only left-right and upper right-lower left motion, how would you move 10 units up? Question. Using only left-right and upper right-lower left motion, how would you move x units to the right and y units up? To get out of the plane, we must give our imaginary “car” the ability to fly. Suppose the car can move in two different (“fundamental”) directions in the plane, as above. Suppose also you can steer it in a third direction which allows you to fly up out of the plane (which means that by putting it 2 in reverse, you can burrow down below the plane also): Suppose the blue and red directions in the plane are perpendicular and that the green direction is drawn at a 45◦ angle to the blue direction. This time we can reach any point in “space” (here, the term space is used, in contrast to line and plane, to refer to our usual “3-dimensional” Euclidean space). Let’s refer to locations in this space in terms of left-right (the blue direction), forward-back (the red direction), and up-down (NOT the green direction!). If we wanted to start a journey and travel 10 units forward, 5 units to the right, and 10 units up, how would we do it? Well, forward and right movement is rather easy. The only way to get upward movement is to travel in the green direction. But this also introduces rightward movement — but we can remove any extra rightward movement with leftward√ movement (by driving in the blue direction in reverse). So let’s move in the green direction 10 2 units. This means that we have moved up out of the√ plane 10 units and also to the right 10 units (remember what Pythagoras says: 102 +102 = (10 2)2). Now we can “adjust” the left-right distance by traveling in reverse 5 units in the blue direction (i.e., units to the left). This only leaves the forward (red) direction to deal with and so√ we go 10 units in the red direction. In fact, we can describe the location of our destination as: 10 2 green, −5 blue, 10 red (notice the use of a negative to indicate that we traveled in the blue direction in reverse). Question. What combination of green, blue, red would move you x units forward, y units to the right, and z units up? Of course the questions given above would be a bit easier if our fundamental directions were each perpendicular to the other (if they were mutually orthogonal). This can be accomplished by making the green direction straight up out of the plane (it can also be accomplished by making the green 3 direction straight down): Now it is easy to see how to go a certain distance forward, to the right, and up. To go x units forward, y units to the right, and z units up, simply go x units in the red direction, y units in the blue direction, and z units in the green direction. These travel plans are even valid if x, y and/or z are negative (just put the car in reverse, as needed). You might also notice that any other direction in this space can be thought of as a combination of these three fundamental directions. Therefore this space is 3-dimensional — we can travel to any point in this space provided we can travel in three fundamental directions. Here, by “fundamental direction” we mean three directions, no one of which is a combination of the other two (i.e., three linearly independent directions). So this story of a flying car with a strange steering mechanism gives us some intuition as to why lines are 1-dimensional, planes are 2-dimensional, and space (as we are familiar with in our experience) is 3-dimensional. Now, we need to think very abstractly to move up another dimension. We want our imaginary car to have the ability to point in a new direction that is independent of left-right, forward-back, and up-down. Of course we cannot visualize such a direction since we live in the usual 3-dimensional space described above and we do not have physical access to a fourth fundamental direction (but we can access it mentally). Just as we moved from the line to the plane by introducing a new direction not on the line, and as we moved from the plane to the 3- dimensional space by introducing a new direction not in the plane, we now move from 3-dimensions to 4-dimensions by introducing a new direction not in our 3-D space. In fact, things seemed easier in the previous conversation when we had fundamental directions which were perpendicular to each other (“mutually orthogonal”), so let’s start with the left-right, forward-back, up-down directions and add a new direction perpendicular to each of these. Since the 4-dimensional space we are moving into is often called hyperspace, let’s call the new direction “hyperup” (when we go hyperup in reverse, we are moving “hyperdown”). Now if we want to go x units forward, y units to the left, z units up, and 0 units hyperup, then we just travel in the manner described above in our usual 3-D space. In this case, our 3-D space is a slice of 4-D hyperspace (the slice we get by going 0 units hyperup). In fact, if we go 1 unit hyperup and then “look around” in the other three fundamental directions, we find another copy of our usual 3-D space — it’s just moved hyperup 1 unit from our familiar copy of 3-D space. In fact, just as you can think of a plane as being produced by taking a bunch of copies of a line, one on top of another (this is where the text “inks up” a line and drags it), and just as you can think of our 3-dimensional space as a bunch of planes lying one on top of another, we can think of 4-D hyperspace as a pile of copies of our usual 3-D space piled hyperup on top (or is it hypertop?) of each other.
Load more

Recommended publications
  • Interaction of the Past of Parallel Universes
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Interaction of the Past of parallel universes Alexander K. Guts Department of Mathematics, Omsk State University 644077 Omsk-77 RUSSIA E-mail: [email protected] October 26, 1999 ABSTRACT We constructed a model of five-dimensional Lorentz manifold with foliation of codimension 1 the leaves of which are four-dimensional space-times. The Past of these space-times can interact in macroscopic scale by means of large quantum fluctuations. Hence, it is possible that our Human History consists of ”somebody else’s” (alien) events. In this article the possibility of interaction of the Past (or Future) in macroscopic scales of space and time of two different universes is analysed. Each universe is considered as four-dimensional space-time V 4, moreover they are imbedded in five- dimensional Lorentz manifold V 5, which shall below name Hyperspace. The space- time V 4 is absolute world of events. Consequently, from formal standpoints any point-event of this manifold V 4, regardless of that we refer it to Past, Present or Future of some observer, is equally available to operate with her.Inotherwords, modern theory of space-time, rising to Minkowsky, postulates absolute eternity of the World of events in the sense that all events exist always. Hence, it is possible interaction of Present with Past and Future as well as Past can interact with Future. Question is only in that as this is realized. The numerous articles about time machine show that our statement on the interaction of Present with Past is not fantasy, but is subject of the scientific study.
    [Show full text]
  • Near-Death Experiences and the Theory of the Extraneuronal Hyperspace
    Near-Death Experiences and the Theory of the Extraneuronal Hyperspace Linz Audain, J.D., Ph.D., M.D. George Washington University The Mandate Corporation, Washington, DC ABSTRACT: It is possible and desirable to supplement the traditional neu rological and metaphysical explanatory models of the near-death experience (NDE) with yet a third type of explanatory model that links the neurological and the metaphysical. I set forth the rudiments of this model, the Theory of the Extraneuronal Hyperspace, with six propositions. I then use this theory to explain three of the pressing issues within NDE scholarship: the veridicality, precognition and "fear-death experience" phenomena. Many scholars who write about near-death experiences (NDEs) are of the opinion that explanatory models of the NDE can be classified into one of two types (Blackmore, 1993; Moody, 1975). One type of explana tory model is the metaphysical or supernatural one. In that model, the events that occur within the NDE, such as the presence of a tunnel, are real events that occur beyond the confines of time and space. In a sec ond type of explanatory model, the traditional model, the events that occur within the NDE are not at all real. Those events are merely the product of neurobiochemical activity that can be explained within the confines of current neurological and psychological theory, for example, as hallucination. In this article, I supplement this dichotomous view of explanatory models of the NDE by proposing yet a third type of explanatory model: the Theory of the Extraneuronal Hyperspace. This theory represents a Linz Audain, J.D., Ph.D., M.D., is a Resident in Internal Medicine at George Washington University, and Chief Executive Officer of The Mandate Corporation.
    [Show full text]
  • Arxiv:1902.02232V1 [Cond-Mat.Mtrl-Sci] 6 Feb 2019
    Hyperspatial optimisation of structures Chris J. Pickard∗ Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom and Advanced Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba, Sendai, 980-8577, Japan (Dated: February 7, 2019) Anticipating the low energy arrangements of atoms in space is an indispensable scientific task. Modern stochastic approaches to searching for these configurations depend on the optimisation of structures to nearby local minima in the energy landscape. In many cases these local minima are relatively high in energy, and inspection reveals that they are trapped, tangled, or otherwise frustrated in their descent to a lower energy configuration. Strategies have been developed which attempt to overcome these traps, such as classical and quantum annealing, basin/minima hopping, evolutionary algorithms and swarm based methods. Random structure search makes no attempt to avoid the local minima, and benefits from a broad and uncorrelated sampling of configuration space. It has been particularly successful in the first principles prediction of unexpected new phases of dense matter. Here it is demonstrated that by starting the structural optimisations in a higher dimensional space, or hyperspace, many of the traps can be avoided, and that the probability of reaching low energy configurations is much enhanced. Excursions into the extra dimensions are progressively eliminated through the application of a growing energetic penalty. This approach is tested on hard cases for random search { clusters, compounds, and covalently bonded networks. The improvements observed are most dramatic for the most difficult ones. Random structure search is shown to be typically accelerated by two orders of magnitude, and more for particularly challenging systems.
    [Show full text]
  • General N-Dimensional Rotations
    General n-Dimensional Rotations Antonio Aguilera Ricardo Pérez-Aguila Centro de Investigación en Tecnologías de Información y Automatización (CENTIA) Universidad de las Américas, Puebla (UDLAP) Ex-Hacienda Santa Catarina Mártir México 72820, Cholula, Puebla [email protected] [email protected] ABSTRACT This paper presents a generalized approach for performing general rotations in the n-Dimensional Euclidean space around any arbitrary (n-2)-Dimensional subspace. It first shows the general matrix representation for the principal n-D rotations. Then, for any desired general n-D rotation, a set of principal n-D rotations is systematically provided, whose composition provides the original desired rotation. We show that this coincides with the well-known 2D and 3D cases, and provide some 4D applications. Keywords Geometric Reasoning, Topological and Geometrical Interrogations, 4D Visualization and Animation. 1. BACKGROUND arctany x x 0 arctany x x 0 The n-Dimensional Translation arctan2(y, x) 2 x 0, y 0 In this work, we will use a simple nD generalization of the 2D and 3D translations. A point 2 x 0, y 0 x (x1 , x2 ,..., xn ) can be translated by a distance vec- In this work we will only use arctan2. tor d (d ,d ,...,d ) and results x (x, x ,..., x ) , 1 2 n 1 2 n 2. INTRODUCTION which, can be computed as x x T (d) , or in its expanded matrix form in homogeneous coordinates, The rotation plane as shown in Eq.1. [Ban92] and [Hol91] have identified that if in the 2D space a rotation is given around a point, and in the x1 x2 xn 1 3D space it is given around a line, then in the 4D space, in analogous way, it must be given around a 1 0 0 0 plane.
    [Show full text]
  • Can a "Hyperspace" Really Exist? by Edward J. Zampino (August, 1997
    Can a "Hyperspace" really Exist? by Edward J. Zampino (August, 1997 ) NASA Lewis Research Center 21000 Brookpark Rd., Cleveland, Ohio 44135 (216)433-2042 [email protected] L._ _ _- Abstract The idea of "hyperspace" is suggested as a possible approach to faster-than-light (FTL) motion. A brief summary of a 1986 study on the Euclidean representation of space-time by the author is presented. Some new calculations on the relativistic momentum and energy of a free particle in Euclidean "hyperspace" are now added and discussed. The superimposed Energy-Momentum curves for subluminal particles, tachyons, and particles in Euclidean "hyperspace" are presented. It is shown that in Euclidean "hyperspace", instead of a relativistic time dilation there is a time "compression" effect. Some fundamental questions are presented. 1. INTRODUCTION In the George Lucas and 20 _hCentury Fox Production of STAR WARS, Han Solo engages a drive mechanism on the Millennium Falcon to attain the "boost-to-light-speed". The star field visible from the bridge streaks backward and out of sight as the ship accelerates to light speed. Science fiction writers have fantasized time and time again the existence of a special space into which a space ship makes a transition. In this special space, objects can exceed the speed of light and then make a transition from the special space back to "normal space". Can a special space (which I will refer to as hyperspace) exist? Our first inclination is to emphatically say nol However, what we have learned from areas of research such as Kaluza-Klein theory[l], Grand Unified supersymmetry theory [2], and Heterotic string theory [3] is that there indeed can be many types of spaces.
    [Show full text]
  • Dimension Theory: Road to the Forth Dimension and Beyond
    Dimension Theory: Road to the Fourth Dimension and Beyond 0-dimension “Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality, for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy.” ― Edwin A. Abbott, Flatland: A Romance of Many Dimensions 0-dimension Space of zero dimensions: A space that has no length, breadth or thickness (no length, height or width). There are zero degrees of freedom. The only “thing” is a point. = ∅ 0-dimension 1-dimension Space of one dimension: A space that has length but no breadth or thickness A straight or curved line. Drag a multitude of zero dimensional points in new (perpendicular) direction Make a “line” of points 1-dimension One degree of freedom: Can only move right/left (forwards/backwards) { }, any point on the number line can be described by one number 1-dimension How to visualize living in 1-dimension Stuck on an endless one-lane one-way road Inhabitants: points and line segments (intervals) Live forever between your front and back neighbor.
    [Show full text]
  • How Spatial Is Hyperspace? Interacting with Hypertext Documents: Cognitive Processes and Concepts
    CYBERPSYCHOLOGY & BEHAVIOR Volume 4, Number 1, 2001 Mary Ann Liebert, Inc. How Spatial Is Hyperspace? Interacting with Hypertext Documents: Cognitive Processes and Concepts PATRICIA M. BOECHLER ABSTRACT The World Wide Web provides us with a widely accessible technology, fast access to massive amounts of information and services, and the opportunity for personal interaction with nu- merous individuals simultaneously. Underlying and influencing all of these activities is our basic conceptualization of this new environment; an environment we can view as having a cognitive component (hyperspace) and a social component (cyberspace). This review argues that cognitive psychologists have a key role to play in the identification and analysis of how the processes of the mind interact with the Web. The body of literature on cognitive processes provides us with knowledge about spatial perceptions, strategies for navigation in space, memory functions and limitations, and the formation of mental representations of environ- ments. Researchers of human cognition can offer established methodologies and conceptual frameworks toward investigation of the cognitions involved in the use of electronic envi- ronments like the Web. INTRODUCTION only a new cognitive environment (“hyper- space”) where information is perceived, stored, OR CENTURIES, TEXT-BASED LITERATURE has manipulated, and retrieved in new ways, but Fbeen the primary mode for disseminating also a new social environment (“cyberspace”), and receiving information. The cognitive where one individual’s cognitions about this processes involved in reading and writing new environment interact with those of other books have been studied extensively for people who share it. Unlike physical space, the decades as they have played a crucial role in World Wide Web is not constrained by dis- the development of humankind.
    [Show full text]
  • The Dual Space
    October 3, 2014 III. THE DUAL SPACE BASED ON NOTES BY RODICA D. COSTIN Contents 1. The dual of a vector space 1 1.1. Linear functionals 1 1.2. The dual space 2 1.3. Dual basis 3 1.4. Linear functionals as covectors; change of basis 4 1.5. Functionals and hyperplanes 5 1.6. Application: Spline interpolation of sampled data (Based on Prof. Gerlach's notes) 5 1.7. The bidual 8 1.8. Orthogonality 8 1.9. The transpose 9 1.10. Eigenvalues of the transpose 11 1. The dual of a vector space 1.1. Linear functionals. Let V be a vector space over the scalar field F = R or C. Recall that linear functionals are particular cases of linear transformation, namely those whose values are in the scalar field (which is a one dimensional vector space): Definition 1. A linear functional on V is a linear transformation with scalar values: φ : V ! F . A simple example is the first component functional: φ(x) = x1. We denote φ(x) ≡ (φ; x). Example in finite dimensions. If M is a row matrix, M = [a1; : : : ; an] n where aj 2 F , then φ : F ! F defined by matrix multiplication, φ(x) = Mx, is a linear functional on F n. These are, essentially, all the linear func- tionals on a finite dimensional vector space. Indeed, the matrix associated to a linear functional φ : V ! F in a basis BV = fv1;:::; vng of V , and the (standard) basis f1g of F is the row vector (1) [φ1; : : : ; φn]; where φj = (φ; vj) 1 2 BASED ON NOTES BY RODICA D.
    [Show full text]
  • Physics of the Hyperspace
    Visionary sciences The need for higher Physics of the dimensions In order to get closer to his target, na- mely to create a unified quantum field theory, Burkhard Heim initially con- centrated on gravitation. According hyperspace to Einstein´s formula E= mc2, every energy has a mass and thus, every en- ergy field is tainted with a so-called Burkhard Heim‘s fieldtheory field mass. Therefore, the gravitatio- nal field of a mass also has some ener- and radionics gy which in turn corresponds to a mass. In turn, a gravitational field emanates More than thirty years ago, the physicist from this field mass. Newton and Ein- stein neglected this field mass in their Burkhard Heim succeeded in conceptualizing the individual gravitational theories and existence of higher dimensions in mathematical could therefore work with relatively formulas. According to Heim, the entire physics of simple approximation formulas. Bur- khard Heim was the first physicist who gravitation is only possible in a twelve-dimensional consistently integrated this field mass hyperspace that mathematically illustrates the in his model. He called the resulting tangible and the mental reality. Marcus Schmieke, gravitational formula a „transcenden- tal equation“ - an equation that has no who was personally introduced by Burkhard Heim generally valid solution that encom- to cosmology, explains Heim‘s field theory and its passes all scales of the universe. One is only able to derive various approxi- consequences for radionics. mation formulas for different magni- tude ranges of the cosmos. By Marcus Schmieke, Berlin. In the macroscopic range of medium distances, the solution of Heim´s tran- scendental equation substantially cor- responds to Newton‘s law of universal he twelve dimensions are divi- vels of existence.
    [Show full text]
  • A Recursive Algorithm for the K-Face Numbers of Wythoffian-N-Polytopes Constructed from Regular Polytopes
    Journal of Information Processing Vol.25 528–536 (Aug. 2017) [DOI: 10.2197/ipsjjip.25.528] Invited Paper A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes Jin Akiyama1,a) Sin Hitotumatu2 Motonaga Ishii3,b) Akihiro Matsuura4,c) Ikuro Sato5,d) Shun Toyoshima6,e) Received: September 6, 2016, Accepted: May 25, 2017 Abstract: In this paper, an n-dimensional polytope is called Wythoffian if it is derived by the Wythoff construction from an n-dimensional regular polytope whose finite reflection group belongs to An,Bn,Cn,F4,G2,H3,H4 or I2(p). Based on combinatorial and topological arguments, we give a matrix-form recursive algorithm that calculates the num- ber of k-faces (k = 0, 1,...,n) of all the Wythoffian-n-polytopes using Schlafli-Wytho¨ ff symbols. The correctness of the algorithm is reconfirmed by the method of exhaustion using a computer. Keywords: Wythoff construction, Wythoffian polytopes, k-face numbers, recursive algorithm, Coxeter group ( c ) volumes and surface areas. 1. Introduction Our purpose has been completed and already implemented High-dimensional polytopes have been studied mainly by the as computer programs. We will introduce a new methodology following methodologies: to study volumes, surface areas, number of k-faces sharing a (1) Combinatorics and Topology, (2) Graph Theory, (3) Group vertex, etc. of Wythoffian-n-polytopes constructed from regular Theory and (4) Metric Geometry. See [1], [2], [8], [9], [10], [11], polytopes in the forthcoming papers. In this paper, as a first [14], [15], [18], [19] for some background researches on tilings, step, we develop a recursive algorithm that computes the f-vector lattices and polytopes.
    [Show full text]
  • A DIMENSION THEORY of the NATURE of MIND ' By1 Wilson M. Van Dusen Thesis Presented to the Faculty of Arts of the University Of
    ^ 000492 A DIMENSION THEORY OF THE NATURE OF MIND ' by1 Wilson M. Van Dusen Thesis presented to the Faculty of Arts of the University of Ottawa through the Institute of Psychology as partial ful­ fillment of the requirements for the degree of Dootor of Philosophy. ^ % - B:3!lOr"t.Q'JrS -rSity o^ ° Ottawa, Canada, 1952 UMI Number: DC53485 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI® UMI Microform DC53485 Copyright 2011 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ACKNOWLEDGMENT This thesis was prepared under the guidance of the Director of the Institute of Psychology, Reverend Father Raymond H. Shevenell, O.M.I. I am also Indebted to Professor Olga Brldgman of the University of California who gave freedom and encourage­ ment in the original development of the idea, to Professor Rheem Jarrett for his guidance and stimulus in mathematics, and to Professor George Adams in cosmology. I am grateful for helpful comments and criticism from my colleagues Ernest Belden and Jack Danlelson of the Neuropsychiatry Consultation Service, Fort Ord, California, and from Gilles Chagnon and Bruce Mendel of the University of Ottawa.
    [Show full text]
  • Arxiv:1204.4952V2 [Math.HO] 17 May 2012
    Sculptures in S3 Saul Schleimer Henry Segerman∗ Mathematics Institute Department of Mathematics and Statistics University of Warwick University of Melbourne Coventry CV4 7AL Parkville VIC 3010 United Kingdom Australia [email protected] [email protected] Abstract We construct a number of sculptures, each based on a geometric design native to the three-dimensional sphere. Using stereographic projection we transfer the design from the three-sphere to ordinary Euclidean space. All of the sculptures are then fabricated by the 3D printing service Shapeways. 1 Introduction The three-sphere, denoted S3, is a three-dimensional analog of the ordinary two-dimensional sphere, S2. In n+ general, the n–dimensional sphere is a subset of Euclidean space, R 1, as follows: n n+1 2 2 2 S = f(x0;x1;:::;xn) 2 R j x0 + x1 + ··· + xn = 1g: Thus S2 can be seen as the usual unit sphere in R3. Visualising objects in dimensions higher than three is non-trivial. However for S3 we can use stereographic projection to reduce the dimension from four to three. n n n Let N = (0;:::;0;1) be the north pole of S . We define stereographic projection r : S − fNg ! R by x0 x1 xn−1 r(x0;x1;:::;xn) = ; ;:::; : 1 − xn 1 − xn 1 − xn See [1, page 27]. Figure 1a displays stereographic projection in dimension one. For any point (x;y) 2 S1 − fNg draw the straight line L through N and (x;y). Then L meets R1 at a single point; this is r(x;y). Notice that the figure is also a two-dimensional cross-section of stereographic projection in any dimension.
    [Show full text]